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C2×C3×D21×C53 #27211331

order13356
isoclinismisoclinic class of D21
automorphism?
homologyAbelianization (H1) = C22×C3×C53 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×D21×C53
unambiguous semidirect product(C2×C3)⋊(C3×D7×C53)

Splittings

Fitting, socleC2×C32×C7×C53 ⋊ C2
derivedC3×C7 ⋊ C22×C3×C53
HallC32×C7×C53 ⋊ C22
HallC32×C53 ⋊ C2×D7
HallC7×C53 ⋊ C2×C3×D3
HallC32×C7 ⋊ C22×C53
SylowC32 ⋊ C2×D7×C53
SylowC7 ⋊ C2×C3×D3×C53

Subgroups

subgroupG/H
center, hypercenterC2×C3×C53D21
commutator, cosocle, hypocenterC3×C7C22×C3×C53
FrattiniC1G
Fitting, socleC2×C32×C7×C53C2
LCSG ▷ C3×C7
UCSC1 ◁ C2×C3×C53

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 4452

A minimal faithful permutation representation — 2 generators on 68 points:

(2,3)(4,5)(6,7)(8,9,10)(12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(1,2,4,6,7,5,3)(11,12,13)(67,68)

Conjugacy classes & character table

3816 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13356, 124);